Dyadic decomposition

Quick description

When estimating a sum or integral of an expression which depends very much on the magnitude of some expression , try decomposing the sum or integral based on the dyadic regions for .This is a special case of the divide and conquer strategy.(Occasionally, one needs also to deal with the exceptional set .)

Prerequisites

Undergraduate calculus.For more geometric integrals, a good geometric intuition is also useful.

Example 1

Example 2

Suppose we want to obtain upper and lower bounds for the integral

on the unit sphere in , where is surface measure, is an exponent, and is a parameter, which we assume to not lie on the unit sphere.For the purposes of this exercise, we are willing to lose multiplicative constants that depend on and , but we are interested in the behavior in (in particular, how does this integral degenerate as ?)

Let's first deal with some easy cases.Suppose is quite small, e.g..Then we see that is comparable to .In this case a simple application of the base times height method gives upper and lower bounds of (up to constants).

Similarly, when is quite large, e.g., then is comparable to , and the base times height method now gives upper and lower bounds of (again up to constants).

Now we turn to the intermediate region .Here, the base times height method does not work so well.Indeed, if we let , this method gives an upper bound of and a lower bound of (up to constants); these two bounds diverge as .The problem here is that the integrand is no longer of the uniformly distributed type that is amenable to the base times height bounds; instead, the integrand is quite large when is nearly parallel to , and quite small when is distant from .

After looking at the geometry of the situation, it becomes clear that the most important parameter controlling the magnitude of the integrand is the angle between and .Indeed, using some trigonometry (and taking advantage of useful simplifications that are available when one is willing to lose constants, e.g.that is comparable to ) we soon see that the distance between and is comparable to .In particular, we see that the "natural" scale for is that of .

Once one sees this, it becomes natural to dyadically decompose with respect to , and specifically into the very near region and dyadic shells for .(It is pointless to decompose the very near region any further, because when , is no longer the dominant influence on the integrand.)Actually, since cannot exceed , we can stop the decomposition once exceeds .

We can then apply the base times height method to each region.In the near region , the integrand has height bounded above and below by (up to constants), and the base has measure (up to constants), leading to a net bound of .Similarly, on each shell , the height is comparable to , and the base is comparable to (except possibly for the very last shell, but let's ignore this), leading to a net bound of .Summing this geometric series, we obtain a final upper or lower bound comparable to when , when , and when .

General discussion

Dyadic decomposition is a fairly efficient technique that generally only "loses a logarithm" or "loses an epsilon exponent" compared to the optimal bound.This is because on many cases, the number of dyadic scales which are "active" or "important" is only logarithmically large with respect to some relevant parameter.For instance, if one is estimating a sum by decomposing into dyadic regions , the number of such regions is only , and so even if one only decides to estimate a single, "worst-case" region rather than handle all the regions at once, one only expects to be off by a factor of at worst.

The only time when dyadic decomposition is truly inefficient is when one expects quite a lot of cancellation between different dyadic regions.In such cases, of course, this sort of decomposition should be avoided.

Sometimes (as one sees in Example 2) one does not want to dyadically decompose the very small or very low values of , thus for instance one could decompose the region into dyadic regions for , but refuse to decompose the region any further.The choice of when to decompose and when not to varies from problem to problem; one has to weigh the benefits of a decomposition against the costs.

For an integral such as , sometimes one wants to dyadically decompose the independent variable , for instance into dyadic annuli .In other cases, it makes more sense to decompose the dependent variable , for instance into level sets .The optimal choice of decomposition is often determined by the geometry of the situation.If, for instance, one is trying to integrate a function which is becoming singular on some set (e.g.a point, a line, a surface, etc.), then dyadically decomposing based on the distance to that set , e.g.to sets

There is nothing sacred about powers of here.Sometimes it is more advantageous to use a sparser dyadic decomposition, e.g.powers of , where is a parameter to be optimized later.

It is often advantageous to decompose using smooth cutoffs rather than sharp cutoffs, especially if there is oscillation present and one wishes to maximize the effect of cancellation.